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2 Fundamenta Informaticae

2 2004

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Understanding Basic Automata Theory in the Continuous Time Setting

100%

Trakhtenbrot B. A.

Fundamenta Informaticae

2004

tom Vol. 62, nr 1

69--121

Paradigms, in which continuous time is involved in cooperation with, or instead of, discrete time appear now in different areas related to automata, logic and interaction. Unfortunately, they are accompanied by a plethora of definitions, terminology and notation, which is not free of ad-hoc and ambiguous decisions. The overuse of definitions from scratch of intricate notions without a previous, explicit core of basic generic notions engenders further models and formalisms, and it is not clear where to stop. Hence (quoting J.Hartmanis), the challenge ``to isolate the right concepts, to formulate the right models, and to discard many others, that do not capture the reality we want to understand...". We undertake this challenge wrt some automata-theoretic concepts and issues that appear in the literature on continuous-time circuits and hybrid automata, by keeping to the following guidelines: 1. Building on Basic Automata Theory. 2. Coherence with original or potential discrete-time paradigms, whose continuous-time analogs and/or mutants we would like to understand. 3. Functions, notably input/output behavior of devices, should not be ignored in favor of sets (languages) accepted by them. The paper outlines the approach which emerged in previous research [PRT, RT, T3, T4, R] and in teaching experience [T0, T2]. As an illustration we offer a precise explanation of the evasive relationship between hybrid automata, constrained automata and control circuits.

Synchronous Circuits over Continuous Time: Feedback Reliability and Completeness

51%

Pardo D. , Rabinovich A. , Trakhtenbrot B. A.

Fundamenta Informaticae

2004

tom Vol. 62, nr 1

123--137

To what mathematical models do digital computer circuits belong? In particular: (i) (Feedback reliability.) Which cyclic circuits should be accepted? In other words, under which conditions is causally faithful the propagation of signals along closed cycles of the circuit? (ii) (Comparative power and completeness.) What are the appropriate primitives upon which circuits may be (or should be) assembled? There are well-known answers to these questions for circuits operating in discrete time, and they point on the exclusive role of the unit-delay primitive. For example: (i) If every cycle in the circuit N passes through a delay, then N is feedback reliable. (ii) Every finite-memory operator F is implementable in a circuit over unit-delay and pointwise boolean gates. In what form, if any, can such phenomena and results be extended to circuits operating in continuous time? This is the main problem considered (and, hopefully, solved to some extent) in this paper. In order to tackle the problems one needs more insight into specific properties of continuous time signals and operators that are not visible at discrete time.